Sequences of 3 Consecutive Integers with Rising Phi

Theorem

The following ordered triples of consecutive integers have $\phi$ values which are strictly increasing:

$105, 106, 107$

$165, 166, 167$

$315, 316, 317$

Proof

\(\ds \map \phi {105}\)

\(=\)

\(\ds 48\)

$\phi$ of $105$

\(\ds \map \phi {106}\)

\(=\)

\(\ds 52\)

$\phi$ of $106$

\(\ds \map \phi {107}\)

\(=\)

\(\ds 106\)

Euler Phi Function of Prime: $107$ is prime

\(\ds \map \phi {165}\)

\(=\)

\(\ds 80\)

$\phi$ of $165$

\(\ds \map \phi {166}\)

\(=\)

\(\ds 82\)

$\phi$ of $166$

\(\ds \map \phi {167}\)

\(=\)

\(\ds 166\)

Euler Phi Function of Prime: $167$ is prime

\(\ds \map \phi {315}\)

\(=\)

\(\ds 144\)

$\phi$ of $315$

\(\ds \map \phi {316}\)

\(=\)

\(\ds 156\)

$\phi$ of $316$

\(\ds \map \phi {317}\)

\(=\)

\(\ds 316\)

Euler Phi Function of Prime: $317$ is prime

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $105$